A cloth of length five and two-third meters was cut into 17 equal pieces. How long is each piece?
So there are two key bits of information that we’ve been given, first of all, that we’ve got a cloth of length five and two-thirds meters. And then we’re also told that it’s cut into 17 equal pieces. And the fact that the pieces are equal in length means that this tells us which one of our operations that we’re going to use. So what we’re going to want to do is divide five and two-thirds by 17. And that’s because, as we said, we’ve got 17 equal pieces.
Well, we’ve got five and two-thirds divided by 17. How are we going to carry out this calculation? Well, what we have on the left-hand side here is a mixed number. And it’s called a mixed number because what we have is both an integer part and fraction part. So our integer is five and our fraction is two-thirds. So the first thing we’re going to want to do is convert our mixed number into an improper or top-heavy fraction. So the way that we do this is we split it into two parts.
So first of all, we take a look at the integer. So we’ve got five units, and what we want to do is convert this into thirds. So we do five multiplied by three. And then this is over three because it’s thirds that we’re trying to convert into. And we can do this because we know that for every unit, there are three-thirds. So therefore, five units, we’re gonna have five multiplied by three-thirds. And then we add on the two-thirds that we already have. Well, because we’ve got the same denominator now, because they’re both thirds, what we can do is just add the numerators. And we had five multiplied by three, which is 15, and then add two, because we had two-thirds, which is gonna give us 17 as our numerator. So therefore, we got 17 over three or seventeen-thirds. So we’ve now converted our mixed number into an improper or top-heavy fraction.
Well, we could’ve completed the same process using a mental method. And to do that, what we’d have done is we’d have multiplied our integer, so our five, by the denominator of our fraction, so that’s three, so that’s 15. Then added the numerator of our fraction, which is two, which is gonna give us 17, and then put the whole thing over the denominator of our fraction, which would be three, which again would give us 17 over three or seventeen-thirds. So that’s a quick mental method you can do if you’re comfortable with this. So now what we’ve got is 17 over three divided by 17. So now what we can do is use one particular method to help us solve this. And to do that, what we do is we put 17 over one. So we have a fraction here as well. And we can do that because 17 is like having 17 over one.
So now what we’ve got is one fraction divided by another fraction. So what we can use is one of our general rules for operations with fractions. And that general rule is our division rule for fractions. What it tells us is that if we’ve got 𝑎 over 𝑏 divided by 𝑐 over 𝑑, then what we do is we flip the second fraction and then multiply our fractions. So we have 𝑎 over 𝑏 multiplied by 𝑑 over 𝑐. And this flipping of a fraction is known as finding the reciprocal of that fraction. And then what we can do is take this one step further and actually show you how you would multiply two fractions. Well multiplying fractions is quite straightforward because all we do is we multiply the numerators and we multiply the denominators. So in our example, we’d get 𝑎𝑑 over 𝑏𝑐.
Okay, great. So now let’s apply this to our question. So therefore, what we’ve got is 17 over three multiplied by one over 17. So now what we can do is cross-cancel or divide the numerator and denominator by 17. So what we’re gonna have is one over three multiplied by one over one. So now what we’re gonna do is multiply the numerators and the denominators. So we’ve got one multiplied by one which is one, three multiplied by one which is three. So we’re gonna be left with one-third or one over three. Now it’s worth noting that we could’ve just known this because what we have is one over three, which is a third, multiplied by, well we’ve got one over one. Well, one over one is just one. So if we have a third multiplied by one, we’d just get a third.
So even though we’d found our answer of a third, we’ve got to look back at the question. And we’re told that a cloth of length five and two-third meters was cut into 17 equal pieces. Well, therefore, we want to say that each piece is one-third of a meter long, so we can’t forget our units there. Well, in fact, we’ve solved this problem using a method which demonstrates how to divide fractions. But could we have solved it any earlier? Well, if we take a look at the second line of working, we could’ve solved it here because what we had was seventeen-thirds or 17 over three divided by 17. Well, if we have 17 of something and we divide it by 17, then we’re just left with one of that thing.
So therefore, if we have seventeen-thirds and we divide it by 17, we would’ve been left with one-third. So that would have been another quicker way of getting to the final answer.